Some things can be defined that cannot exist, such as "A square circle in two dimensional space" or "2+2=1" --and some things can be described that do not exist but could have existed or might come to exist (unicorns). And, I suggest, that there may be indefinitely many things that exist for which we do not have any successful definition. "Consciousness" might be a candidate, insofar as some philosophers are right in thinking we may never have a good or at-least problem-free definition.
As an aside, your question raises the need for a good definition of definitions. I will not attempt such a philosophy of definitions here, but you might check out the Stanford Encyclopedia entries bearing on philosophy of language for further, useful material. Paradoxically, if nothing can exist than cannot be defined, and we have no definition of being defined, we all might be in trouble.
Thinking further: I suspect you may be principally concerned with the problem of affirming that something (X) exists, and whether this affirmation is meaningful if we lack a definition of X. On the face of it, there would be a problem with someone claiming: "Call the reporters. There is something I will refer to as 'N,' but I have absolutely no idea or definition of what 'N' might be. It could be an animal or number or time of day, for I know." Such a claim would be as bizarre as what we find in Alice in Wonderland. Even so, I suggest that we should distinguish claims about meaningful speech and claims about what does or does not exist. Even if we cannot make claims about what does or does not exist without (at least vague) definitions, it is another thing to claim that there only exists things we can make meaningful claims about. Sadly, we can imagine the whole human species perishing from some force which we cannot comprehend (and thus we cannot define) That is such a grim thought to end this reply, let me change the example: we can imagine that cancer and depression might be eradicated by a force that we human beings cannot comprehend or define.